A COMPREHENSIVE STUDY ON THE VARIATIONAL FORMULATION OF THE FRACTAL-FRACTIONAL KDV-ZK EQUATION WITH VISCOELASTICITY INSIGHTS AND APPLICATIONS

In the field of nonlinear equation research, especially for solitary wave equations, obtaining exact solutions is extremely challenging. However, variational theory offers crucial insights. This paper focuses on the fractal-fractional Korteweg-de Vries-Zakharov-Kuznetsov (KdV-ZK) equation with viscoelasticity and conducts a comprehensive and in-depth study. By using the semi-inverse method to carefully construct the variational formulation, setting the trial functional in the energy integral ingeniously, and determining the unknown function according to the Euler–Lagrange equation, the variational principle of this equation is successfully obtained. In this process, a new modified equation form is derived with the help of the potential function. Further in-depth exploration reveals that under specific conditions, the equation can be transformed into an ordinary differential equation. Through detailed analysis, the conditions for the existence of periodic solutions are obtained, and effective methods such as He’s frequency formulation or the homotopy perturbation method are used to determine the frequency–amplitude relationship. Meanwhile, the variational formulation obtained in this paper has important application value in numerical simulation and analytical analysis. It not only opens up new directions for the energy method research of nonlinear wave equations but also provides valuable references and guidance for theoretical and applied research in related fields, effectively promoting the development of this field.